1 /*- 2 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG> 3 * Copyright (c) 2017 Mahdi Mokhtari <mmokhi@FreeBSD.org> 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28 /* 29 * The algorithm is very close to that in "Implementing the complex arcsine 30 * and arccosine functions using exception handling" by T. E. Hull, Thomas F. 31 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on 32 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335, 33 * http://dl.acm.org/citation.cfm?id=275324. 34 * 35 * See catrig.c for complete comments. 36 * 37 * XXX comments were removed automatically, and even short ones on the right 38 * of statements were removed (all of them), contrary to normal style. Only 39 * a few comments on the right of declarations remain. 40 */ 41 42 #include <sys/cdefs.h> 43 #include <complex.h> 44 #include <float.h> 45 46 #include "invtrig.h" 47 #include "math.h" 48 #include "math_private.h" 49 50 #undef isinf 51 #define isinf(x) (fabsl(x) == INFINITY) 52 #undef isnan 53 #define isnan(x) ((x) != (x)) 54 #define raise_inexact() do { volatile float junk __unused = 1 + tiny; } while(0) 55 #undef signbit 56 #define signbit(x) (__builtin_signbitl(x)) 57 58 #if LDBL_MAX_EXP != 0x4000 59 #error "Unsupported long double format" 60 #endif 61 62 static const long double 63 A_crossover = 10, 64 B_crossover = 0.6417, 65 FOUR_SQRT_MIN = 0x1p-8189L, 66 HALF_MAX = 0x1p16383L, 67 QUARTER_SQRT_MAX = 0x1p8189L, 68 RECIP_EPSILON = 1 / LDBL_EPSILON, 69 SQRT_MIN = 0x1p-8191L; 70 71 #if LDBL_MANT_DIG == 64 72 static const union IEEEl2bits 73 um_e = LD80C(0xadf85458a2bb4a9b, 1, 2.71828182845904523536e+0L), 74 um_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L); 75 #define m_e um_e.e 76 #define m_ln2 um_ln2.e 77 static const long double 78 /* The next 2 literals for non-i386. Misrounding them on i386 is harmless. */ 79 SQRT_3_EPSILON = 5.70316273435758915310e-10, /* 0x9cc470a0490973e8.0p-94 */ 80 SQRT_6_EPSILON = 8.06549008734932771664e-10; /* 0xddb3d742c265539e.0p-94 */ 81 #elif LDBL_MANT_DIG == 113 82 static const long double 83 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */ 84 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */ 85 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */ 86 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17; /* 0x13988e1409212e7d0321914321a55.0p-167 */ 87 #else 88 #error "Unsupported long double format" 89 #endif 90 91 static const volatile float 92 tiny = 0x1p-100; 93 94 static long double complex clog_for_large_values(long double complex z); 95 96 static inline long double 97 f(long double a, long double b, long double hypot_a_b) 98 { 99 if (b < 0) 100 return ((hypot_a_b - b) / 2); 101 if (b == 0) 102 return (a / 2); 103 return (a * a / (hypot_a_b + b) / 2); 104 } 105 106 static inline void 107 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable, 108 long double *B, long double *sqrt_A2my2, long double *new_y) 109 { 110 long double R, S, A; 111 long double Am1, Amy; 112 113 R = hypotl(x, y + 1); 114 S = hypotl(x, y - 1); 115 116 A = (R + S) / 2; 117 if (A < 1) 118 A = 1; 119 120 if (A < A_crossover) { 121 if (y == 1 && x < LDBL_EPSILON * LDBL_EPSILON / 128) { 122 *rx = sqrtl(x); 123 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) { 124 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S); 125 *rx = log1pl(Am1 + sqrtl(Am1 * (A + 1))); 126 } else if (y < 1) { 127 *rx = x / sqrtl((1 - y) * (1 + y)); 128 } else { 129 *rx = log1pl((y - 1) + sqrtl((y - 1) * (y + 1))); 130 } 131 } else { 132 *rx = logl(A + sqrtl(A * A - 1)); 133 } 134 135 *new_y = y; 136 137 if (y < FOUR_SQRT_MIN) { 138 *B_is_usable = 0; 139 *sqrt_A2my2 = A * (2 / LDBL_EPSILON); 140 *new_y = y * (2 / LDBL_EPSILON); 141 return; 142 } 143 144 *B = y / A; 145 *B_is_usable = 1; 146 147 if (*B > B_crossover) { 148 *B_is_usable = 0; 149 if (y == 1 && x < LDBL_EPSILON / 128) { 150 *sqrt_A2my2 = sqrtl(x) * sqrtl((A + y) / 2); 151 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) { 152 Amy = f(x, y + 1, R) + f(x, y - 1, S); 153 *sqrt_A2my2 = sqrtl(Amy * (A + y)); 154 } else if (y > 1) { 155 *sqrt_A2my2 = x * (4 / LDBL_EPSILON / LDBL_EPSILON) * y / 156 sqrtl((y + 1) * (y - 1)); 157 *new_y = y * (4 / LDBL_EPSILON / LDBL_EPSILON); 158 } else { 159 *sqrt_A2my2 = sqrtl((1 - y) * (1 + y)); 160 } 161 } 162 } 163 164 long double complex 165 casinhl(long double complex z) 166 { 167 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 168 int B_is_usable; 169 long double complex w; 170 171 x = creall(z); 172 y = cimagl(z); 173 ax = fabsl(x); 174 ay = fabsl(y); 175 176 if (isnan(x) || isnan(y)) { 177 if (isinf(x)) 178 return (CMPLXL(x, y + y)); 179 if (isinf(y)) 180 return (CMPLXL(y, x + x)); 181 if (y == 0) 182 return (CMPLXL(x + x, y)); 183 return (CMPLXL(nan_mix(x, y), nan_mix(x, y))); 184 } 185 186 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 187 if (signbit(x) == 0) 188 w = clog_for_large_values(z) + m_ln2; 189 else 190 w = clog_for_large_values(-z) + m_ln2; 191 return (CMPLXL(copysignl(creall(w), x), 192 copysignl(cimagl(w), y))); 193 } 194 195 if (x == 0 && y == 0) 196 return (z); 197 198 raise_inexact(); 199 200 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 201 return (z); 202 203 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 204 if (B_is_usable) 205 ry = asinl(B); 206 else 207 ry = atan2l(new_y, sqrt_A2my2); 208 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 209 } 210 211 long double complex 212 casinl(long double complex z) 213 { 214 long double complex w; 215 216 w = casinhl(CMPLXL(cimagl(z), creall(z))); 217 return (CMPLXL(cimagl(w), creall(w))); 218 } 219 220 long double complex 221 cacosl(long double complex z) 222 { 223 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 224 int sx, sy; 225 int B_is_usable; 226 long double complex w; 227 228 x = creall(z); 229 y = cimagl(z); 230 sx = signbit(x); 231 sy = signbit(y); 232 ax = fabsl(x); 233 ay = fabsl(y); 234 235 if (isnan(x) || isnan(y)) { 236 if (isinf(x)) 237 return (CMPLXL(y + y, -INFINITY)); 238 if (isinf(y)) 239 return (CMPLXL(x + x, -y)); 240 if (x == 0) 241 return (CMPLXL(pio2_hi + pio2_lo, y + y)); 242 return (CMPLXL(nan_mix(x, y), nan_mix(x, y))); 243 } 244 245 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 246 w = clog_for_large_values(z); 247 rx = fabsl(cimagl(w)); 248 ry = creall(w) + m_ln2; 249 if (sy == 0) 250 ry = -ry; 251 return (CMPLXL(rx, ry)); 252 } 253 254 if (x == 1 && y == 0) 255 return (CMPLXL(0, -y)); 256 257 raise_inexact(); 258 259 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 260 return (CMPLXL(pio2_hi - (x - pio2_lo), -y)); 261 262 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 263 if (B_is_usable) { 264 if (sx == 0) 265 rx = acosl(B); 266 else 267 rx = acosl(-B); 268 } else { 269 if (sx == 0) 270 rx = atan2l(sqrt_A2mx2, new_x); 271 else 272 rx = atan2l(sqrt_A2mx2, -new_x); 273 } 274 if (sy == 0) 275 ry = -ry; 276 return (CMPLXL(rx, ry)); 277 } 278 279 long double complex 280 cacoshl(long double complex z) 281 { 282 long double complex w; 283 long double rx, ry; 284 285 w = cacosl(z); 286 rx = creall(w); 287 ry = cimagl(w); 288 if (isnan(rx) && isnan(ry)) 289 return (CMPLXL(ry, rx)); 290 if (isnan(rx)) 291 return (CMPLXL(fabsl(ry), rx)); 292 if (isnan(ry)) 293 return (CMPLXL(ry, ry)); 294 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z)))); 295 } 296 297 static long double complex 298 clog_for_large_values(long double complex z) 299 { 300 long double x, y; 301 long double ax, ay, t; 302 303 x = creall(z); 304 y = cimagl(z); 305 ax = fabsl(x); 306 ay = fabsl(y); 307 if (ax < ay) { 308 t = ax; 309 ax = ay; 310 ay = t; 311 } 312 313 if (ax > HALF_MAX) 314 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1, 315 atan2l(y, x))); 316 317 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 318 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x))); 319 320 return (CMPLXL(logl(ax * ax + ay * ay) / 2, atan2l(y, x))); 321 } 322 323 static inline long double 324 sum_squares(long double x, long double y) 325 { 326 327 if (y < SQRT_MIN) 328 return (x * x); 329 330 return (x * x + y * y); 331 } 332 333 static inline long double 334 real_part_reciprocal(long double x, long double y) 335 { 336 long double scale; 337 uint16_t hx, hy; 338 int16_t ix, iy; 339 340 GET_LDBL_EXPSIGN(hx, x); 341 ix = hx & 0x7fff; 342 GET_LDBL_EXPSIGN(hy, y); 343 iy = hy & 0x7fff; 344 #define BIAS (LDBL_MAX_EXP - 1) 345 #define CUTOFF (LDBL_MANT_DIG / 2 + 1) 346 if (ix - iy >= CUTOFF || isinf(x)) 347 return (1 / x); 348 if (iy - ix >= CUTOFF) 349 return (x / y / y); 350 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF) 351 return (x / (x * x + y * y)); 352 scale = 1; 353 SET_LDBL_EXPSIGN(scale, 0x7fff - ix); 354 x *= scale; 355 y *= scale; 356 return (x / (x * x + y * y) * scale); 357 } 358 359 long double complex 360 catanhl(long double complex z) 361 { 362 long double x, y, ax, ay, rx, ry; 363 364 x = creall(z); 365 y = cimagl(z); 366 ax = fabsl(x); 367 ay = fabsl(y); 368 369 if (y == 0 && ax <= 1) 370 return (CMPLXL(atanhl(x), y)); 371 372 if (x == 0) 373 return (CMPLXL(x, atanl(y))); 374 375 if (isnan(x) || isnan(y)) { 376 if (isinf(x)) 377 return (CMPLXL(copysignl(0, x), y + y)); 378 if (isinf(y)) 379 return (CMPLXL(copysignl(0, x), 380 copysignl(pio2_hi + pio2_lo, y))); 381 return (CMPLXL(nan_mix(x, y), nan_mix(x, y))); 382 } 383 384 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 385 return (CMPLXL(real_part_reciprocal(x, y), 386 copysignl(pio2_hi + pio2_lo, y))); 387 388 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) { 389 raise_inexact(); 390 return (z); 391 } 392 393 if (ax == 1 && ay < LDBL_EPSILON) 394 rx = (m_ln2 - logl(ay)) / 2; 395 else 396 rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4; 397 398 if (ax == 1) 399 ry = atan2l(2, -ay) / 2; 400 else if (ay < LDBL_EPSILON) 401 ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2; 402 else 403 ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2; 404 405 return (CMPLXL(copysignl(rx, x), copysignl(ry, y))); 406 } 407 408 long double complex 409 catanl(long double complex z) 410 { 411 long double complex w; 412 413 w = catanhl(CMPLXL(cimagl(z), creall(z))); 414 return (CMPLXL(cimagl(w), creall(w))); 415 } 416